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Advanced calculus

Name: Advanced calculus
ISBN: 0-86720-122-3

By Lynn H. Loomis

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Book Description

Table of Contents

Title Page
Preface
Contents
Chapter 0: Introduction
- 0.1 Logic: Quantifiers
- 0.2 The Logical Connectives
- 0.3 Negations of Quantifiers
- 0.4 Sets
- 0.5 Restricted Variables
- 0.6 Ordered Pairs and Relations
- 0.7 Functions and Mappings
- 0.8 Product Sets; Index Notation
- 0.9 Composition
- 0.10 Duality
- 0.11 The Boolean Operations
- 0.12 Partitions and Equivalence Relations
Chapter 1: Vector Spaces
- 1.1 Fundamental Notions
- 1.2 Vector Spaces and Geometry
- 1.3 Product Spaces and Hom(V,W)
- 1.4 Affine Subspaces and Quotient Spaces
- 1.5 Direct Sums
- 1.6 Bilinearity
Chapter 2: Finite-Dimensional Vector Spaces
- 2.1 Bases
- 2.2 Dimension
- 2.3 The Dual Space
- 2.4 Matrices
- 2.5 Trace and Determinant
- 2.6 Matrix Computations
- 2.7* The Diagonalization of a Quadratic Form
Chapter 3: The Differential Calculus
- 3.1 Review in R
- 3.2 Norms
- 3.3 Continuity
- 3.4 Equivalent Norms
- 3.5 Infinitesimals
- 3.6 The Differential
- 3.7 Directional Derivatives; the Mean-Value Theorem
- 3.8 The Differential and Product Spaces
- 3.9 The Differential and R^n
- 3.10 Elementary Applications
- 3.11 The Implicit-Function Theorem
- 3.12 Submanifolds and Lagrange Multipliers
- 3.13* Functional Dependence
- 3.14* Uniform Continuity and Function-Valued Mappings
- 3.15* The Calculus of Variations
- 3.16* The Second Differential and the Classification of Critical Points
- 3.17* Higher Order Differentials. The Taylor Formula
Chapter 4: Compactness and Completeness
- 4.1 Metric Spaces; Open and Closed Sets
- 4.2* Topology
- 4.3 Sequential Convergence
- 4.4 Sequential Compactness
- 4.5 Compactness and Uniformity
- 4.6 Equicontinuity
- 4.7 Completeness
- 4.8 A First Look at Banach Algebras
- 4.9 The Contraction Mapping Fixed-Point Theorem
- 4.10 The Integral of a Parameterized Arc
- 4.11 The Complex Number System
- 4.12* Weak Methods
Chapter 5: Scalar Product Spaces
- 5.1 Scalar Products
- 5.2 Orthogonal Projection
- 5.3 Self-Adjoint Transformations
- 5.4 Orthogonal Transformations
- 5.5 Compact Transformations
Chapter 6: Differential Equations
- 6.1 The Fundamental Theorem
- 6.2 Differentiable Dependence on Parameters
- 6.3 The Linear Equation
- 6.4 The nth-Order Linear Equation
- 6.5 Solving the Inhomogeneous Equation
- 6.6 The Boundary-Value Problem
- 6.7 Fourier Series
Chapter 7: Multilinear Functionals
- 7.1 Bilinear Functionals
- 7.2 Multilinear Functionals
- 7.3 Permutations
- 7.4 The Sign of a Permutation
- 7.5 The Subspace A^n of Alternating Tensors
- 7.6 The Determinant
- 7.7 The Exterior Algebra
- 7.8 Exterior Powers of Scalar Product Spaces
- 7.9 The Star Operator
Chapter 8: Integration
- 8.1 Introduction
- 8.2 Axioms
- 8.3 Rectangles and Paved Sets
- 8.4 The Minimal Theory
- 8.5 The Minimal Theory (Continued)
- 8.6 Contented Sets
- 8.7 When is a Set Contented?
- 8.8 Behavior Under Linear Distortions
- 8.9 Axioms for Integration
- 8.10 Integration of Contented Functions
- 8.11 The Change of Variables Formula
- 8.12 Successive Integration
- 8.13 Absolutely Integrable Functions
- 8.14 Problem Set: The Fourier Transform
Chapter 9: Differentiable Manifolds
- 9.1 Atlases
- 9.2 Functions, Convergence
- 9.3 Differentiable Manifolds
- 9.4 The Tangent Space
- 9.5 Flows and Vector Fields
- 9.6 Lie Derivatives
- 9.7 Linear Differential Forms
- 9.8 Computations with Coordinates
- 9.9 Riemann Metrics
Chapter 10: The Integral Calculus on Manifolds
- 10.1 Compactness
- 10.2 Partitions of Unity
- 10.3 Densities
- 10.4 Volume Density of a Riemann Metric
- 10.5 Pullback and Lie Derivatives of Densities
- 10.6 The Divergence Theorem
- 10.7 More Complicated Domains
Chapter 11: Exterior Calculus
- 11.1 Exterior Differential Forms
- 11.2 Oriented Manifolds and the Integration of Exterior Differential Forms
- 11.3 The Operator d
- 11.4 Stokes' Theorem
- 11.5 Some Illustrations of Stokes' Theorem
- 11.6 The Lie Derivative of a Differential Form
- Appendix I: "Vector Analysis"
- Appendix II: Elementary Differential Geometry of Surfaces in E^3
Chapter 12: Potential Theory in E^n
- 12.1 Solid Angle
- 12.2 Green's Formulas
- 12.3 The Maximum Principle
- 12.4 Green's Functions
- 12.5 The Poisson Integral Formula
- 12.6 Consquences of the Poisson Integral Formula
- 12.7 Harnack's Theorem
- 12.8 Subharmonic Functions
- 12.9 Dirichlet's Problem
- 12.10 Behavior Near the Boundary
- 12.11 Dirchlet's Principle
- 12.12 Physical Applications
- 12.13 Problem Set: The Calculus of Residues
Chapter 13: Classical Mechanics
- 13.1 The Tangent and Cotangent Bundles
- 13.2 Equations of Variation
- 13.3 The Fundamental Linear Differential Form on T*(M)
- 13.4 The Fundamental Exterior Two-Form on T*(M)
- 13.5 Hamiltonian Mechanics
- 13.6 The Central-Force Problem
- 13.7 The Two-Body Problem
- 13.8 Lagrange's Equations
- 13.9 Variational Principles
- 13.10 Geodesic Coordinates
- 13.11 Euler's Equations
- 13.12 Rigid Body Motion
- 13.13 Small Oscillations
- 13.14 Small Osccillations (Continued)
- 13.15 Canonical Transformations
Selected References
Notation Index
Index